Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{10}{6x(3x + 8)} \div \dfrac{-8}{3x + 8} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{10}{6x(3x + 8)} \times \dfrac{3x + 8}{-8} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 10 \times (3x + 8) } { 6x(3x + 8) \times -8 } $ $ p = \dfrac {10 (3x + 8)} {-8 \times 6x(3x + 8)} $ $ p = \dfrac{10(3x + 8)}{-48x(3x + 8)} $ We can cancel the $3x + 8$ so long as $3x + 8 \neq 0$ Therefore $x \neq -\dfrac{8}{3}$ $p = \dfrac{10 \cancel{(3x + 8})}{-48x \cancel{(3x + 8)}} = -\dfrac{10}{48x} = -\dfrac{5}{24x} $